Problem: Divide the following complex numbers: $\dfrac{7(\cos(\frac{1}{2}\pi) + i \sin(\frac{1}{2}\pi))}{\cos(\pi) + i \sin(\pi)}$ (The dividend is plotted in blue and the divisor in plotted in green. Your current answer will be plotted orange.)
Answer: Dividing complex numbers in polar forms can be done by dividing the radii and subtracting the angles. The first number ( $7(\cos(\frac{1}{2}\pi) + i \sin(\frac{1}{2}\pi))$ ) has angle $\frac{1}{2}\pi$ and radius 7. The second number ( $\cos(\pi) + i \sin(\pi)$ ) has angle $1\pi$ and radius 1. The radius of the result will be $\frac{7}{1}$ , which is 7. The difference of the angles is $\frac{1}{2}\pi - 1\pi = -\frac{1}{2}\pi$ The angle $-\frac{1}{2}\pi$ is negative. A complex number goes a full circle if its angle is increased by $2 \pi$ , so it goes back to itself. Because of that, angles of complex numbers are convenient to keep between $0$ and $2 \pi$ $-\frac{1}{2}\pi + 2 \pi = \frac{3}{2}\pi$ The radius of the result is $7$ and the angle of the result is $\frac{3}{2}\pi$.